Here we consider three circles c1, c2, c3 belonging to the same bundle of circles I(c1, c2), generated by the first two of them. We draw tangents from points A of circle c1 to c2, c3, and consider the tangency chord BC. Find the equation of the envelope of lines BC. The same question when c1, c2, c3 are conics.
It seems to include formidable calculations. The case in which the two inner circles c2, c3 coincide with one circle (c'(M,r), with M(a,0)), and c1 centered at the origin (c1: x2 + y2 = R2) is easier. The equation in this case is that of a conic, and takes the form Ax2 + 2Bx + Cy2 + D = 0, where
A = R2 + a2, B = a(r2-a2-R2), C = R2, D = (R2a2 + r4 + 2r2a3 + a4).
![[alogo]](alogo.png){kind=small}
Envelope of tangency chords ![[0_0]](TangentsEnvelope_files/TangentsEnvelope_0_0_0.png)
![[0_1]](TangentsEnvelope_files/TangentsEnvelope_0_0_1.png)
![[0_2]](TangentsEnvelope_files/TangentsEnvelope_0_0_2.png)
![[1_0]](TangentsEnvelope_files/TangentsEnvelope_0_1_0.png)
![[1_1]](TangentsEnvelope_files/TangentsEnvelope_0_1_1.png)
![[1_2]](TangentsEnvelope_files/TangentsEnvelope_0_1_2.png)